Strong Efficiency For Vector Set-Valued Optimization Problems

When both the objective function and constrained function are arc wise connected cone-convex functions, with directional derivative and alternative theorem, the necessary conditions are obtained for constrained vector-valued optimization problem to obtain its strongly efficient solutions. By using scalarization theorem for the strong efficient point, Kuhn-Tucker sufficient optimality condition is obtained for vector-valued optimization problem to obtain its strongly efficient solutions. And the strong-subgradient for set-valued map is introduced, whose existence theorem is proved. An equivalent depiction of the strong-subgradient is presented. As applications, the sufficient and necessary conditions for the set-valued optimization problem with constraint to attain its strongly efficient elements are given.